1 Ideal Opamp Circuits
1.1 The Voltage Follower
We redraw the circuit of non-inverting amplier as shown in Fig.(1).
+
vo
vi
Ri
RF
Figure 1: The non-inverting amplier
If we force either RF = 0 or Ri = we obtain vo = vi. In practice both actions areperformed simultaneously to obtain the voltage follower circuit as shown in Fig.(2).
+vi
vo
Figure 2: The voltage follower or buer amplier
The voltage follower is so called as the output voltage follows the input withabsolute faithfulness. Besides, it eliminates the presence of loading - it tends toseparate the load from the source. It behaves like an voltage controlled voltage source(VCVS). VCVS is another name for voltage buer and it job is to isolate the eectsof source and load. VCVS has the following properties: Ri = and Ro = 0 Fig.(3)claries the necessity for VCVS.
Any practical voltage source can be modelled as an ideal voltage source with aseries resistance Rs coming along with it. If it were directly connected to load thevoltage experienced by the load is:
vL =RL
Rs +RLvi (1)
It is less than the intended voltage vi. Now, we will insert the VCVS or buer asseen in Fig.(4) to see the isolation process:
1
+vi
Rs
RL
vL
Figure 3: A practical voltage source connected to load.
+vi
Rs
RL
vL
+
Figure 4: Source-load isolation using voltage follower.
Since the input impedance is innity for an ideal opamp, no current ows throughRs and hence no voltage loss occurs across it. Hence, we have v+ = vi. The voltagefollower conguration makes the output equal to vL = v+ = vi. The voltage followermakes the practical voltage source appear as though it were an ideal voltage sourcenullifying the eect of Rs. Table (1) shows the properties of other controlled sourceswhich we will see later.
Controlled Source Ri RoVCVS 0VCCS CCVS 0 0CCCS 0
Table 1: Properties of controlled sources
Next, we see some applications of opamp. In all these circuits the opamp isembedded in negative feedback loop.
2
1.2 Inverting Summing Amplifier
We will now turn our attention to an important amplier called as the Summer orSumming amplier whose circuit is shown in Fig.(5).
+
v1
v2 vo
R1
R2
RF
Figure 5: The summing amplier
+
v1
v2 vo
R1
R2
RF
i1
i2
iF
Figure 6: Summing Amplier Analysis
From Fig.(6), since v is at virtual ground we have,
i1 =v1R1
(2)
i2 =v2R2
(3)
iF = i1 + i2 (4)
=v1R1
+v2R2
(5)
voRF
=v1R1
+v2R2
(6)
vo = (RFR1
v1 +RFR2
v2
)(7)
3
The output voltage vo is the weighted sum of inputs v1 and v2 with weightsRFR1
and RFR2
, respectively. When the special condition RF = R1 = R2 holds then,
vo = (v1 + v2) (8)The output is the negative of the sum of the input voltages and hence this circuit
is called inverting summing amplier. The circuit can be extended easily for three ormore inputs.
1.3 Non-Inverting Summing Amplifier
Fig.(7) shows the non-inverting counterpart of the circuit shown in previous section.
vo
RF
+
Ri
R1
R2
v1
v2
Figure 7: Non-Inverting Summing Amplier
Now, this circuit appears a little tricky for analysis. However, we can use sourcesuperposition and analyze by using one source at a time. If v2 is shorted and v1 isonly present (as seen in Fig.(8)), then the analysis proceeds as follows:
v+ =R2
R1 +R2v1 (9)
v = v+ =R2
R1 +R2v1 (10)
The voltage v+ experiences the same gain as that of non-inverting amplier ofgain 1 + RF
Ri. That is the partial output due to v1 is,
vo1 =
(1 +
RFRi
)R2
R1 +R2v1. (11)
Likewise, the partial output due to the input v2 is:
vo2 =
(1 +
RFRi
)R1
R1 +R2v2 (12)
4
vo
RF
+
Ri
R1
R2
v1
Figure 8: Non-Inverting Summing Amplier Analysis
The total output is,
vo = vo1 + vo2 =
(1 +
RFRi
)v1R2 + v2R1R1 +R2
. (13)
In the special condition that all resistors are equal R1 = R2 = Ri = RF = R wehave,
vo = (1 + 1)v1 + v21 + 1
= v1 + v2 (14)
and is rightly called the Non-Inverting Summing Amplier. Once again this pro-cedure can be extended to three or more inputs.
1.4 Subtractor
Fig.(9) shows the subtractor circuit with all resistors equal.Once again we use source superposition and apply each source induvidually. When
v1 is only present we short circuit v2. The resulting circuit is shown in Fig.(10) In thiscircumstance v+ = 0V as no current ows into opamp due to high input impedance.The circuit behaves as though it is an Inverting Amplier with the partial outputbeing,
vo1 = RRv1 = v1 (15)
When we switch roles, this time with v1 shorted, the circuit is identical to anNon-Inverting Amplier with gain 1 + R
R= 2, however we also note that v+ =
v22
because of voltage divider action (see Fig.(11)). Hence,
5
+
R
R
R
Rv1
v2
vo
Figure 9: Opamp Subtractor Circuit
+
R
R
v1vo
R/2
Figure 10: Subtractor with Partial input v1
vo2 =
(1 +
R
R
)v22
= v2. (16)
Finally, the total output is vo = vo1 + vo2 = v2 v1, hence the name subtractor.
1.5 Excercises
1. Design a circuit that simply adds three voltages.
6
+
R
R
vov2
R
R
v2/2
Figure 11: Subtractor with partial input v2
7
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